Questions: Five hundred consumers are surveyed about a new brand of snack food, Crunchicles. Their age groups and preferences are given in the table. 18-24 25-34 35-55 55 and over Total Liked Crunchicles 1 17 18 75 111 Disliked Crunchicles 20 27 13 30 90 No Preference 69 19 43 168 299 Total 90 63 74 273 500 One consumer from the survey is selected at random. Leave all answers in a reduced fraction. a. What is the probability that the consumer is 18-24 years of age, given that he/she dislikes Crunchicles? b. What is the probability that the selected consumer dislikes Crunchicles? c. What is the probability that the selected consumer is 35-55 years old or likes Crunchicles? d. If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles?

Solution Steps

a. To find the probability that the consumer is 18-24 years of age given that they dislike Crunchicles, use the conditional probability formula: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). Here, \( A \) is the event that the consumer is 18-24, and \( B \) is the event that the consumer dislikes Crunchicles.

b. The probability that the selected consumer dislikes Crunchicles is the number of consumers who dislike Crunchicles divided by the total number of consumers surveyed.

c. To find the probability that the selected consumer is 35-55 years old or likes Crunchicles, use the formula for the probability of the union of two events: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Here, \( A \) is the event that the consumer is 35-55 years old, and \( B \) is the event that the consumer likes Crunchicles.

To find \( P(A|B) \), where \( A \) is the event that the consumer is 18-24 years old and \( B \) is the event that the consumer dislikes Crunchicles, we use the formula:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{20}{90} = \frac{2}{9} \]

The probability that a randomly selected consumer dislikes Crunchicles is given by:

\[ P(B) = \frac{90}{500} = \frac{9}{50} \]

To find \( P(A \cup B) \), where \( A \) is the event that the consumer is 35-55 years old and \( B \) is the event that the consumer likes Crunchicles, we use the formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Calculating each component:

• \( P(A) = \frac{74}{500} = \frac{37}{250} \)
• \( P(B) = \frac{111}{500} \)
• \( P(A \cap B) = \frac{18}{500} = \frac{9}{250} \)

Now substituting these values into the formula:

\[ P(A \cup B) = \frac{37}{250} + \frac{111}{500} - \frac{9}{250} \]

Converting \( \frac{111}{500} \) to a common denominator of 500:

\[ P(A \cup B) = \frac{74}{500} + \frac{111}{500} - \frac{18}{500} = \frac{167}{500} \]

Final Answer